You're right that rich people probably aren't going to be hanging receipts next to their original paintings, but they certainly will bring up the fact that it's an original at every possible opportunity. The "original-ness" is definitely where all the value is coming from, whether from owning the original painting, or the "original" NFT.
But you still paid a few million more for the original painting, even though it's virtually indistinguishable from a replica. It's more helpful to think of an original painting as a bundle of two things:
Original painting = thing that looks like a painting + bragging rights
An NFT is just the "bragging rights" part of the original painting, and the JPG that anyone can access is the "thing that looks like a painting ".
And what's the "innate value" of an original painting, compared to a forgery? They look identical to the naked eye, and can probably fool a fair chunk of experts as well. Sure, you might be able to tell the difference using some advanced analytical methods (eg. carbon-14 dating or whatever), but does the slightly lower carbon-14 isotope content justify the massive premium?